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Honors Geometry Section 5.2 Areas of Triangles and Quadrilaterals WHY? RHL E F You can see that the area of parallelogram ABCD is equal to the area of rectangle EBCF. For a parallelogram with base b and height h, the area is given by the formula: A parallelogram = ______ b h Note that the height is the length of the segment perpendicular to the base from a point on the opposite side which is called the altitude of the parallelogram. 2s s 3 4 3 s + A 15 4 3 60 3 u 2 A 10 6 60 8 x 60 x 60 / 8 7.5 u Any triangle is half of a parallelogram. For a triangle with base b and height h, the area is given by the formula: 1 bh A triangle = ________ 2 The height is the length of the ____________ altitude to the base Example: Find the area of to the nearest 1000th. AC sin 25 10 AC 4.226 BC cos 25 10 BC 9.063 A .5 4.226 9.063 19.150 u 2 Example: A triangle has an area of 56 and a base of 10. Find its height. A 1 bh 2 56 1 10h 2 h 11.2 A .5(10)(3) 15 u 2 Trigonometry and the Area of a Triangle Using your knowledge of trigonometry, express h in terms of sinC. h sin C b b sin C h 1 Substituting this into the formula A bh , and using a as the 2 base we get 1 A ab sin C 2 We have just discovered that the area of a triangle can be expressed using the lengths of two sides and the sine of the included angle. Example: Use what you have learned above to find the area of parallelogram ABCD to the nearest 1000th. A// gram 2( Atriangle ) A// gram 1 2 15 25 sin 50 2 A// gram 15 25 sin 50 A// gram 287.267cm 2 An altitude of a trapezoid is a segment perpendicular to the two bases with an endpoint in each of the bases. The length of an altitude will be the height of the trapezoid. 1 b2 h 2 1 b1h 2 1 h(b2 b1 ) 2 1 b2h 1 b1h 2 2 For a trapezoid with bases b1 and b2 and height h, the area of a trapezoid is given by the formula: 1 Atrap. 2 hb1 b2 Recall that the diagonals of both rhombuses and kites are perpendicular. 1 1 Akite 1 BC AE 2 BC DE 2 E BC AE 1 BC DE 2 2 Akite 1 BC AE DE 2 1 Akite Ar hombus d1d 2 2